Medieval castle stairs were often built to ascend in narrow, clockwise spirals so right-handed castle defenders could use their swords more easily. This design put those on the way up at a disadvantage (unless they were left-handed). The steps were also uneven to give defenders the advantage of anticipating each step’s size while attackers tripped over them. SourceSource 2Source 3

Not really the best illustration since it totally negates the effect by having a wide open space for those ascending. Castle tower staircases tended to look like this:

Extremely tight quarters, with a central supporting pillar that is very, very thoroughly in the way of your right arm.

Wider, less steep designs tend to come later once castles moved away from being fortresses to simply noble family homes with the advent of gunpowder.

Oh! Pre-gunpowder military tactics are my jam! I don’t know why, but this is one of my favorite little details about defensive fortifications, because the majority handedness of attackers isn’t usually something you think about when studying historical wars. But strategically-placed walls were used basically worldwide as a strategy to secure gates and passages against advancing attackers, because most of the world’s population is right-handed (and has been since the Stone Age).

Pre-Columbian towns near the Mississippi and on the East coast did this too. They usually surrounded their towns with palisades, and they would build the entrance to the palisade wall in a zigzag – always with the wall to the right as you entered, to hinder attackers and give an advantage to the defender. Here’s some gates with some examples of what I’m talking about:

Notice that, with the exception of the last four (which are instead designed to congregate the attackers in a space so they can be picked off by archers, either in bastions or on the walls themselves) and the screened gate (which, in addition to being baffled, also forces the attackers to defend their flank) all of these gates are designed with central architectural idea that it’s really hard to kill someone with a wall in your way.

In every culture in the world, someone thought to themselves, “Hey it’s hard to swing a weapon with a wall on your right-hand side,” and then specifically built fortifications so that the attackers would always have the wall on their right. And I think that’s really neat.

Ooh, ooh, also: Bodiam Castle in Sussex used to have a right-angled bridge so any attacking forces would be exposed to archery fire from the north-west tower on their right side (ie: sword in the right hand, shield on the useless left side):

These tactics worked so well for so long because until quite recently lefties got short shrift and had it trained (if they were lucky) or beaten out of them.

Use of sword and shield is a classic demonstration of how right-handedness predominated. There’s historical mention of left-handed swordsmen (gladiators and Vikings), and what a problem they were for their opponents, but that only applies to single combat.

A left-handed hoplite or housecarl simply couldn’t fight as part of a phalanx or shield wall, since the shields were a mutual defence (the right side of the shield covered its owner’s left side, its left side covered the right side of his neighbour to the left, and so on down the line) and wearing one on the wrong arm threw the whole tactic out of whack.

Jousting, whether with or without an Italian-style tilt barrier, was run shield-side to shield-side with the lance at a slant (except for the Scharfrennen, a highly specialised style that’s AFAIK unique.) Consequently left-handed knights were physically unable to joust.

The construction of plate armour, whether specialised tournament kit or less elaborate battle gear, is noticeably “right-handed“ - so even if a wealthy knight had his built “left-handed” it would be a waste of time and money; he would still be a square peg in a world of round holes and none of the other kids would play with him.

Even after shields and full armour were no longer an essential part of military equipment, right-hand use was still enforced until quite recently, and to important people as well as ordinary ones - it happened to George VI, father of the present Queen of England. Most swords with complex hilts, such as swept-hilt rapiers and some styles of basket-hilt broadsword, are assymetrical and constructed for right handers. Here’s my schiavona…

It can be held left-handed, but using it with the proper thumb-ring grip, and getting maximum protection from the basket, is right-handed only. (More here.) Some historical examples of left-hand hilts do exist, but they’re rare, and fencing masters had the same “learn to use your right hand” bias as tourney organisers, teachers and almost everyone else. Right-handers were dextrous, but left-handers were sinister, etc., etc.

However, several
predominantly left-handed
families did turn their handedness into advantage, among them the Kerrs / Carrs, a notorious Reiver family along the England-Scotland Borders, by building their fortress
staircases with a spiral the other way to the OP image.

This would seem to be a bad idea, since the attackers (coming upstairs) no longer have their right arms cramped against the centre pillar - however it worked in the Kerrs’ favour because they were used to this mirror-image of reality while nobody else was, and the defender retreating up the spiral had that pillar guarding his right side, while the attacker had to reach out around it…

For the most part Reiver swords weren’t elaborate swept-hilt rapiers but workmanlike basket-hilts. Some from Continental Europe have the handedness of my schiavona with thumb-rings and assymmetrical baskets, but the native “British Baskethilt” is a variant of the Highland claymore* and like it seems completely symmetrical, without even a thumb-ring, which gives equal protection to whichever hand is using it.

*I’m aware there are those who insist “claymore” refers only to two-handers, however the Gaelic term claidheamh-mòr
- “big sword” -
just refers to size, not to a specific type of sword in the way “schiavona” or “karabela” or even “katana” does.

While the two-hander was the biggest sword in common use it was the
claidheamh-mòr; after it dropped out of fashion and the basket-hilt became the biggest sword in common use, that became the
claidheamh-mòr.

When Highlanders in the 1745 Rebellion referred to their basket-hilts as claymores, they obviously gave no thought to the confusion they would create for later compilers of catalogues…

Well if left-handed swordfighting was good enough for the Hero of Time…